We propose a direct estimation method for R\'{e}nyi and f-divergence measures based on a new graph theoretical interpretation. Suppose that we are given two sample sets $X$ and $Y$, respectively with $N$ and $M$ samples, where $\eta:=M/N$ is a constant value. Considering the $k$-nearest neighbor ($k$-NN) graph of $Y$ in the joint data set $(X,Y)$, we show that the average powered ratio of the number of $X$ points to the number of $Y$ points among all $k$-NN points is proportional to R\'{e}nyi divergence of $X$ and $Y$ densities. A similar method can also be used to estimate f-divergence measures. We derive bias and variance rates, and show that for the class of $\gamma$-H\"{o}lder smooth functions, the estimator achieves the MSE rate of $O(N^{-2\gamma/(\gamma+d)})$. Furthermore, by using a weighted ensemble estimation technique, for density functions with continuous and bounded derivatives of up to the order $d$, and some extra conditions at the support set boundary, we derive an ensemble estimator that achieves the parametric MSE rate of $O(1/N)$. Our estimators are more computationally tractable than other competing estimators, which makes them appealing in many practical applications.

提出了新的图论解释下的直接估计方法，用于估计Renyi和f-divergence的度量。通过对Y中k-NN方案点和X中点数之间的平均功率进行估计，可以获得两个密度之间的Renyi divergence估计值，并且这种方法可以用于估计f-divergence度量。通过使用加权合成估计技术，该方法可以用于具有连续和有界导数的密度函数的估计，其能够获得参数MSE率O(1/N)。

使用最近邻比率直接估计信息差异